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  <div class="section" id="numpy-random-randomstate-lognormal">
<h1>numpy.random.RandomState.lognormal<a class="headerlink" href="#numpy-random-randomstate-lognormal" title="Permalink to this headline">¶</a></h1>
<p>method</p>
<dl class="method">
<dt id="numpy.random.RandomState.lognormal">
<code class="sig-prename descclassname">RandomState.</code><code class="sig-name descname">lognormal</code><span class="sig-paren">(</span><em class="sig-param">mean=0.0</em>, <em class="sig-param">sigma=1.0</em>, <em class="sig-param">size=None</em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.RandomState.lognormal" title="Permalink to this definition">¶</a></dt>
<dd><p>Draw samples from a log-normal distribution.</p>
<p>Draw samples from a log-normal distribution with specified mean,
standard deviation, and array shape.  Note that the mean and standard
deviation are not the values for the distribution itself, but of the
underlying normal distribution it is derived from.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the <code class="docutils literal notranslate"><span class="pre">lognormal</span></code> method of a <code class="docutils literal notranslate"><span class="pre">default_rng()</span></code>
instance instead; see <em class="xref py py-obj">random-quick-start</em>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>mean</strong><span class="classifier">float or array_like of floats, optional</span></dt><dd><p>Mean value of the underlying normal distribution. Default is 0.</p>
</dd>
<dt><strong>sigma</strong><span class="classifier">float or array_like of floats, optional</span></dt><dd><p>Standard deviation of the underlying normal distribution. Must be
non-negative. Default is 1.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape.  If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn.  If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <code class="docutils literal notranslate"><span class="pre">mean</span></code> and <code class="docutils literal notranslate"><span class="pre">sigma</span></code> are both scalars.
Otherwise, <code class="docutils literal notranslate"><span class="pre">np.broadcast(mean,</span> <span class="pre">sigma).size</span></code> samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized log-normal distribution.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html#scipy.stats.lognorm" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.stats.lognorm</span></code></a></dt><dd><p>probability density function, distribution, cumulative density function, etc.</p>
</dd>
<dt><a class="reference internal" href="numpy.random.Generator.lognormal.html#numpy.random.Generator.lognormal" title="numpy.random.Generator.lognormal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator.lognormal</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>A variable <em class="xref py py-obj">x</em> has a log-normal distribution if <em class="xref py py-obj">log(x)</em> is normally
distributed.  The probability density function for the log-normal
distribution is:</p>
<div class="math">
<p><img src="../../../_images/math/c3caf36f576cd8982b5d7adb216fc47f2fa23a91.svg" alt="p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}"/></p>
</div><p>where <img class="math" src="../../../_images/math/4a3598141469c2555591e66606a1b86d4ec6dca9.svg" alt="\mu"/> is the mean and <img class="math" src="../../../_images/math/b52df27bfb0b1e3af0c2c68a7b9da459178c2a7d.svg" alt="\sigma"/> is the standard
deviation of the normally distributed logarithm of the variable.
A log-normal distribution results if a random variable is the <em>product</em>
of a large number of independent, identically-distributed variables in
the same way that a normal distribution results if the variable is the
<em>sum</em> of a large number of independent, identically-distributed
variables.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r780462472f1b-1"><span class="brackets">1</span></dt>
<dd><p>Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal
Distributions across the Sciences: Keys and Clues,”
BioScience, Vol. 51, No. 5, May, 2001.
<a class="reference external" href="https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf">https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf</a></p>
</dd>
<dt class="label" id="r780462472f1b-2"><span class="brackets">2</span></dt>
<dd><p>Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme
Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>Draw samples from the distribution:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span> <span class="o">=</span> <span class="mf">3.</span><span class="p">,</span> <span class="mf">1.</span> <span class="c1"># mean and standard deviation</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">lognormal</span><span class="p">(</span><span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
</pre></div>
</div>
<p>Display the histogram of the samples, along with
the probability density function:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">count</span><span class="p">,</span> <span class="n">bins</span><span class="p">,</span> <span class="n">ignored</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">align</span><span class="o">=</span><span class="s1">&#39;mid&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">bins</span><span class="p">),</span> <span class="nb">max</span><span class="p">(</span><span class="n">bins</span><span class="p">),</span> <span class="mi">10000</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">pdf</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">... </span>       <span class="o">/</span> <span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;tight&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<div class="figure align-default">
<img alt="../../../_images/numpy-random-RandomState-lognormal-1_00_00.png" src="../../../_images/numpy-random-RandomState-lognormal-1_00_00.png" />
</div>
<p>Demonstrate that taking the products of random samples from a uniform
distribution can be fit well by a log-normal probability density
function.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># Generate a thousand samples: each is the product of 100 random</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># values, drawn from a normal distribution.</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="p">[]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
<span class="gp">... </span>   <span class="n">a</span> <span class="o">=</span> <span class="mf">10.</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
<span class="gp">... </span>   <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="c1"># scale values to be positive</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">count</span><span class="p">,</span> <span class="n">bins</span><span class="p">,</span> <span class="n">ignored</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">align</span><span class="o">=</span><span class="s1">&#39;mid&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">sigma</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">bins</span><span class="p">),</span> <span class="nb">max</span><span class="p">(</span><span class="n">bins</span><span class="p">),</span> <span class="mi">10000</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">pdf</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">... </span>       <span class="o">/</span> <span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<div class="figure align-default">
<img alt="../../../_images/numpy-random-RandomState-lognormal-1_01_00.png" src="../../../_images/numpy-random-RandomState-lognormal-1_01_00.png" />
</div>
</dd></dl>

</div>


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